Group #20241322 — subgroups

Subgroups — #20241322

Order 6016.

134 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 41 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15688849	C₂	1	2	yes	maximal
2	C₄₇×#391	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
4	C₂²×Q₈×C₄₇	C₄	1	1	yes	characteristic
5	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
6	C₂×C₄²×C₄₇	C₄	1	1	yes	characteristic
7	#95×C₄₇	—	2	2	no	—
8	C₂×Q₈×C₄₇	—	2	1	no	—
9	C₂×Q₈×C₄₇	—	4	1	no	—
10	C₂×Q₈×C₄₇	—	4	1	no	—
11	C₂×Q₈×C₄₇	C₈	1	2	yes	complemented
12	C₂²×C₄×C₄₇	—	2	1	no	—
13	C₂²×C₄×C₄₇	—	2	1	no	—
14	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₄₇	D₄	1	2	yes	—
16	C₄⋊C₄×C₄₇	—	4	1	no	—
17	C₄²×C₄₇	—	4	1	no	—
18	C₂×C₄₇⋊C₈	—	4	2	no	—
19	C₂³×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
20	Q₈×C₄₇	—	4	2	no	—
21	Q₈×C₄₇	—	4	2	no	—
22	C₂×C₄×C₄₇	—	2	1	no	—
23	C₂×C₄×C₄₇	—	2	1	no	—
24	C₂×C₄×C₄₇	—	2	2	no	—
25	C₂×C₄×C₄₇	—	2	2	no	—
26	C₂×C₄×C₄₇	—	2	2	no	—
27	C₂×C₄×C₄₇	—	4	1	no	—
28	C₂×C₄×C₄₇	—	4	1	no	—
29	C₂×C₄×C₄₇	—	4	1	no	—
30	C₂×C₄×C₄₇	—	4	1	no	—
31	C₂×C₄×C₄₇	—	4	2	no	—
32	C₂×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
33	C₂×C₄×C₄₇	M₁₆	1	1	yes	characteristic
34	C₄₇⋊C₈	—	8	2	no	—
35	C₂²×C₄₇	—	2	1	no	—
36	C₂²×C₄₇	—	2	1	no	—
37	C₂²×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
38	C₂²×C₄₇	#99	1	1	yes	characteristic
39	C₂²×C₄₇	#100	1	1	yes	characteristic
40	C₄×C₄₇	—	2	1	no	—
41	C₄×C₄₇	—	2	1	no	—
42	C₄×C₄₇	—	4	2	no	—
43	C₄×C₄₇	—	4	2	no	—
44	C₄×C₄₇	—	4	2	no	—
45	#1319	—	47	1	no	maximal, Sylow
46	C₂×C₄₇	—	2	1	no	—
47	C₂×C₄₇	—	2	1	no	—
48	C₂×C₄₇	#323	1	1	yes	characteristic
49	C₂×C₄₇	#354	1	1	yes	characteristic
50	C₂×C₄₇	#356	1	1	yes	characteristic
51	#391	D₄₇	1	1	yes	characteristic
52	#324	—	47	2	no	—
53	C₄₇	#1319	1	1	yes	characteristic, complemented, Sylow
54	C₂²×Q₈	Dic₄₇	1	1	yes	characteristic
55	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
56	C₂×C₄²	Dic₄₇	1	1	yes	characteristic
57	C₂²⋊C₈	—	94	2	no	—
58	#95	—	2	2	no	—
59	C₂×Q₈	—	2	1	no	—
60	C₂×Q₈	—	4	1	no	—
61	C₂×Q₈	—	4	1	no	—
62	C₂×Q₈	C₄₇⋊C₈	1	2	yes	complemented
63	C₂²×C₄	—	2	1	no	—
64	C₂²×C₄	—	2	1	no	—
65	C₂²×C₄	C₂×Dic₄₇	1	1	yes	characteristic
66	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
67	C₂×C₈	—	188	2	no	—
68	C₄⋊C₄	—	4	1	no	—
69	C₄²	—	4	1	no	—
70	C₂³	C₂²⋊Dic₄₇	1	1	yes	characteristic
71	Q₈	—	4	2	no	—
72	Q₈	—	4	2	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	2	2	no	—
78	C₂×C₄	—	4	1	no	—
79	C₂×C₄	—	4	1	no	—
80	C₂×C₄	—	4	1	no	—
81	C₂×C₄	—	4	1	no	—
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
84	C₂×C₄	#10625329	1	1	yes	characteristic
85	C₈	—	376	2	no	—
86	C₂²	—	2	1	no	—
87	C₂²	—	2	1	no	—
88	C₂²	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
89	C₂²	#13093939	1	1	yes	characteristic
90	C₂²	#13093941	1	1	yes	characteristic
91	C₄	—	2	1	no	—
92	C₄	—	2	1	no	—
93	C₄	—	4	2	no	—
94	C₄	—	4	2	no	—
95	C₄	—	4	2	no	—
96	C₂	—	2	1	no	—
97	C₂	—	2	1	no	—
98	C₂	#15688847	1	1	yes	characteristic
99	C₂	#15688918	1	1	yes	characteristic
100	C₂	#15688922	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15688849

C₂

yes

maximal

C₄₇×#391

C₂

yes

maximal, characteristic

C₂²⋊(C₄₇⋊C₈)

—

C₂²×Q₈×C₄₇

C₄

yes

characteristic

C₂×C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₂×C₄²×C₄₇

C₄

yes

characteristic

#95×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

C₈

yes

complemented

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂²⋊C₄

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₈

yes

characteristic

C₂×C₄×C₄₇

M₁₆

yes

characteristic

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂²⋊C₈

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

#1319

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

yes

characteristic

#324

—

C₄₇

#1319

yes

characteristic, complemented, Sylow

C₂²×Q₈

Dic₄₇

yes

characteristic

C₂×C₄⋊C₄

C₂×D₄₇

yes

characteristic

C₂×C₄²

Dic₄₇

yes

characteristic

C₂²⋊C₈

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₄₇⋊C₈

yes

complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₄₇

yes

characteristic

C₂²×C₄

C₂²⋊D₄₇

yes

—

C₂×C₈

—

188

—

C₄⋊C₄

—

C₄²

—

C₂³

C₂²⋊Dic₄₇

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄₇⋊C₈

yes

characteristic

C₂×C₄

#10625329

yes

characteristic

C₈

—

376

—

C₂²

—

C₂²

—

C₂²

C₂²⋊(C₄₇⋊C₈)

yes

C₂²

yes

C₂²

yes

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₂

—

C₂

—

C₂

yes

C₂

yes

100

C₂

#15688922

yes

characteristic

	subgroup	G/H
center	C₂²	#13093939
commutator	C₂×C₄×C₄₇	C₂×C₈
Frattini	C₂×C₄⋊C₄	C₂×D₄₇
Fitting	C₄₇×#391	C₂
socle	C₂²×C₄₇	#99
hypercenter	#391	D₄₇
hypocenter	C₄₇	#1319
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄⋊C₄ ◁ #391

subgroup

G/H

center

C₂²

#13093939

commutator

C₂×C₄×C₄₇

C₂×C₈

Frattini

C₂×C₄⋊C₄

C₂×D₄₇

Fitting

C₄₇×#391

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄⋊C₄ ◁ #391

N	H ≅ G/N	type
11. C₂×Q₈×C₄₇	C₈	semidirect
53. C₄₇	#1319	semidirect
62. C₂×Q₈	C₄₇⋊C₈	semidirect

H ≅ G/N

type

11. C₂×Q₈×C₄₇

C₈

semidirect

53. C₄₇

#1319

semidirect

62. C₂×Q₈

C₄₇⋊C₈

semidirect

Subgroups — #20241322

Conjugacy classes of subgroups

Complements